One important method of analysing inhomogeneous samples, such as multiple thin layers on a substrate, involves exposing a specimen to a beam of electrons and measuring the emitted x-ray spectrum. Provided the electron beam is sufficiently energetic to penetrate through the layers and reach the substrate, then characteristic x-rays are generated from elements in both the substrate and the various layers and these contribute to the total x-ray spectrum seen by an x-ray detector.
FIG. 1 shows a typical situation where a 10 keV electron beam is incident on a layered sample with three layers of different thickness on a substrate. Many electron trajectories are shown and x-rays may be generated at any point along the electron trajectory as a result of ionisation of atoms. X-rays are emitted in all directions and if an x-ray detector is positioned above the sample, then x-rays emitted towards the detector will provide signals representative of the elements present in the regions excited by the electron beam. In the arrangement of FIG. 1, x-rays will emerge from all three layers and from the substrate. In FIG. 2, the same sample is exposed to a lower energy electron beam. In this case, the electrons only penetrate into the top two layers so there will be no signal from the lowest layer and the substrate. In general, if a series of x-ray spectra are acquired at different incident electron beam energies, typically between 2 keV and 20 keV, then the corresponding spectra will exhibit characteristic x-ray peaks that vary according to the thicknesses and the compositions of the various layers, and indeed the substrate.
Given a particular electron beam energy, the characteristic intensity for a particular element within a multi-layered sample can be expressed as a “k-ratio”. A “k-ratio” is the ratio of the x-ray intensity received for a particular element from the structure (counts per second recorded from a characteristic x-ray emission series such as K, L or M) to that obtained from a flat bulk specimen of pure element under the same experimental conditions. As the name suggests, a k-ratio is dimensionless. Taking this ratio avoids having to know the x-ray detector collection efficiency as a function of energy. By measuring a series of k-ratios, it is sometimes possible to deduce the thicknesses and compositions of the various layers in a multi-layer specimen. Whether this is possible or not depends upon the x ray data, the extent to which information is known about the sample and experimental conditions, and the number of unknowns.
If NE elements in total occur in one or more of the layers or in the substrate and there are NL layers with thickness T1, T2 . . . TNL, and layer L contains concentration CLj of element j, and the substrate contains concentration CSj of element j then the predicted k-ratio for element i at incident electron beam energy E0 can be written as Equation 1 below:—ki=fi(E0, T1, T2, . . . TNL, C11, C12, . . . C1NE, C21, C22 . . . C2 NE, . . . CNL 1, CNL 2, . . . CNL NE, CS1, CS2, . . . CS NE)   Equation 1:where fi is a non-linear function of the layer thicknesses and the compositions of the layers and substrate. Several equations of this form cover the measured element intensities at this one beam energy E0.
There may be more than one equation for a particular element if measurements are made on more than one x-ray emission series for this element (e.g. K or L or M emissions). Measurements may also be taken at further values of the beam energy and in general there will be M of these non-linear equations for k-ratios. The function in Equation 1 typically involves integrations and non-linear functions and in general it is not possible to “invert” the set of equations and write down a formula that expresses the thickness or compositions for any one layer in terms of a set of measured k-ratios. Therefore, to determine a set of thicknesses and compositions (“layer variables”) from a set of x-ray measurements, it is known to use a modelling approach where the parameters of the model are adjusted to find a set of ki that are a “best-fit” to the measured k-ratios (see for example, Chapter 15 in “Numerical Recipes in C”, Second Edition, W.H. Press et al, Cambridge University Press 1999). Thus, a computer program is used to make iterative guesses at the thickness and composition of the layers to find a set that is a close fit with the k-ratios measured from x-ray spectra (see for example, J. L. Pouchou. “X-ray microanalysis of stratified specimens”, Analytica Chimica Acta, 283 (1993), 81-97. This procedure has been made available commercially in the software product “Stratagem” by SAMx, France). The computer program will make a test at each iteration to see if the guesses are not changing significantly between iterations, in which case “convergence” is achieved. Unfortunately in some cases, it is impossible to find a best-fit set of thicknesses and compositions because the measured k-ratios do not reveal enough differences in intensity to resolve the source of the individual contributions to x-ray intensity. In this case, the computer program iterations will fail to converge on a unique solution. Such problems can sometimes be resolved by choosing different x-ray emission series, different beam energies or constraining the range of possible solutions by providing information on some of the thicknesses or compositions where this is known beforehand.
Although there are some guidelines for the choice of beam energies and x-ray series (see for example, J. L. Pouchou, “X-ray Microanalysis of Thin Films and Coatings”, Microchim. Acta 138, 133-152 (2002)), in general it is difficult to prove that a given type of sample can always be analysed successfully by this technique except by extensive experimentation in the hands of an expert.
A further practical difficulty arises in obtaining measured k-ratios. With a given beam current incident on the multilayered specimen, the x-ray intensity for an elemental line is measured by recording x-ray counts in a known time interval. A pure element standard is then placed under the beam and the x-ray intensity is measured. The ratio of the two intensities is the measured k-ratio. If a pure element standard is not available for the element in question, then a compound standard could be used or a pure standard from a different element. In that case, a correction is required to convert the measured intensity to that which would have been obtained from a pure standard of the element in question. Besides the inconvenience of having to have both specimen and standard accessible on the same specimen stage, a critical requirement is that the beam current and incident electron energy must be identical for the measurements on the specimen and standard. In some specialised instruments, it is possible to obtain a direct measure of incident electron beam current and this can be used in principle to make a correction if a different current is used for specimen and standard measurement provided the beam current measuring equipment is correctly calibrated. However, in general, analysis instruments are not provided with apparatus which is able to accurately measure the beam current since such apparatus is costly and also takes time to stabilize.
For the conventional case of analysis of a homogeneous bulk material, it is well known that analysis is possible without the use of standards or beam current measurements, provided all unknown elements emit lines that can be measured and the total concentration is assumed to be 100%. However, as pointed out previously (J. L. Pouchou. “X-ray microanalysis of stratified specimens”, Analytica Chimica Acta, 283 (1993), 81-97): “Contrary to the case of conventional analysis of homogeneous microvolumes, it is not easy in the case of stratified specimens to work with no standard at all, because this would require to know very accurately the beam current and the solid angle of detection.” Furthermore, in some electron beam instruments such as cold field emission scanning electron microscopes, beam current fluctuations make comparative measurements difficult (e.g. R. Gauvin, “Quantitative X-Ray Microanalysis of Heterogeneous Materials Using Monte Carlo Simulations”, Microchim Acta 155, 75-81 (2006)).
Thus, there are two undesirable problems with the electron-excited x-ray analysis of inhomogeneous materials such as multilayered materials: the difficulty of determining the feasibility of performing me required analysis and the difficulty of measuring k-ratios using the same equivalent beam current. The problem of determining feasibility has been addressed in our previous patent application, the contents of which are incorporated herein by reference in their entirety (WO2007/132243).
One technique for overcoming the beam current difficulty for some specific situations has been described where measurements of film and substrate element line intensities are used to form a ratio which is independent of beam current (Raynald Gauvin, Quantitative X-Ray Microanalysis of Heterogeneous Materials Using Monte Carlo Simulations, Microchim Acta 155, 75-81 (2006)). This ratio can be used in a calibration curve method to determine either the thickness of a single pure element film on dissimilar pure element substrate or the diameter of a pure element sphere on a dissimilar pure element substrate. However, since the ratio involves different element x-ray lines, the detector efficiency does not cancel as it does for a k-ratio and a calibration factor has to be determined for each application.